- 30 Nov '02 18:43let N be a positive interger less than 10^2002. when the digit 1 is

placed after the last digit of N, the number formed is three times the

number formed when the digit 1 is placed infront of N. How many

different values of N are there???

but it's even harder without a calculator...i had to do this for a maths

challenge in school, without a calculator-argh!!!

mr genius - 01 Dec '02 12:49
let N be a positive interger less than 10^2002. when the digit 1 is

I think the answer is 333, and I did not use a calculator, just pen and paper. Here's my

placed after the last digit of N, the number formed is three times the

number formed when the digit 1 is placed infront of N. How many

different values of N are there???

but it's even harder without a calculator...i had to do this for a maths

challenge in school, witho...

reasoning:

3(10^x + N) = 10N + 1

=> 7N = 3*10^x - 1

So 3*10^x -1 must be divisible by 7.

As it turns out, 3*10^x - 1 is divisible by 7 if and only if x is of the form 6y - 1. I worked

this out by calculating 3*10^n modulo 7 for n = 1 to 5; 10^6 = 1 (mod 7) (by Fermat's Little

Theorem ) so x can be considered modulo 6. Hence the result, since for n=1 to 5, only

3*10^5 = 1 (mod 7).

Anyway, there are 333 such numbers <= 2002 (3*10^2003 -1 > 7*10^2002, so that's not

allowed.)

If you've haven't done modular arithmetic, what it means for two numbers to be congruent

(mod n) is that they differ only by a multiple of n. This is pretty nifty, as it means you can

throw away multiples of n all over the place. - 02 Dec '02 16:41nope-we haven't done the modular stuff. i did the question only really

with algebra, but with the same reasoning. also, wasn't fermat's last

therom proved right, in that a^n+b^n <> c^n if a,b,c & n are positive

intergers, and n is greater than 2? therefore, if it's doesn't work, what

can it be used for???

G - 02 Dec '02 17:04Genius, Acolyte used something called Fermat's Little Theorem, not

Fermat's Last Theorem.

The Little Theorem (discovered/created? in 1640) is far more useful

than the more well-known Last Theorem. It is often used (amongst

many other things) for testing the primality of numbers. However,

because the proof goes only one way it is what's called 'necessary but

not sufficient'. This means that it holds for all primes, but (sadly!)

also holds for some composite numbers too.

Fermat's Little Theorem says that if n is a prime number, and

(a,n) = 1, then a^(n-1) = 1 (mod n).

Modular/Modulo arithmetic (and some of the results contained

therein) is an extraordinarily powerful idea, reducing calculations with

huuuuge numbers to quite basic problems. It really is an astonishing

tool.

To find out more on Modular arithmetic (there are loads of sites

dealing with it) you could do worse than going to:-

http://www.cut-the-knot.com/blue/Modulo.shtml

To see the proof of Fermat's Little Theorem visit:-

http://www.utm.edu/research/primes/notes/proofs/FermatsLittleTheore

m.html

Mark